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A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n is odd, and of 5^(n/2)*(x^(4/5)*d/dx)^n when n is even.

%I A223171 #28 Dec 25 2023 18:02:28
%S A223171 1,1,5,6,5,6,60,25,66,110,25,66,990,825,125,1056,2640,1200,125,1056,
%T A223171 21120,26400,8000,625,22176,73920,50400,10500,625,22176,554400,924000,
%U A223171 420000,65625,3125,576576,2402400,2184000,682500,81250,3125,576576,17297280
%N A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n is odd, and of 5^(n/2)*(x^(4/5)*d/dx)^n when n is even.
%H A223171 U. N. Katugampola, <a href="http://authors.elsevier.com/a/1QhUNLvMg0Zs~">Mellin Transforms of Generalized Fractional Integrals and Derivatives</a>, Appl. Math. Comput. 257(2015) 566-580.
%H A223171 U. N. Katugampola, <a href="http://arxiv.org/abs/1411.5229">Existence and Uniqueness results for a class of Generalized Fractional Differential Equations</a>, arXiv preprint arXiv:1411.5229, 2014
%e A223171 Triangle begins:
%e A223171 1;
%e A223171 1, 5;
%e A223171 6, 5;
%e A223171 6, 60, 25;
%e A223171 66, 110, 25;
%e A223171 66, 990, 825, 125;
%e A223171 1056, 2640, 1200, 125;
%e A223171 1056, 21120, 26400, 8000, 625;
%e A223171 22176, 73920, 50400, 10500, 625;
%e A223171 22176, 554400, 924000, 420000, 65625, 3125;
%e A223171 576576, 2402400, 2184000, 682500, 81250, 3125;
%e A223171 576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625;
%e A223171 17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625;
%p A223171 a[0]:= f(x):
%p A223171 for i from 1 to 13 do
%p A223171 a[i] := simplify(5^((i+1)mod 2)*x^((3((i+1)mod 2)+1)/5)*(diff(a[i-1],x$1 )));
%p A223171 end do;
%Y A223171 Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K A223171 nonn,tabf
%O A223171 0,3
%A A223171 _Udita Katugampola_, Mar 20 2013